Conjugacy-separable implies residually finite

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., conjugacy-separable group) must also satisfy the second group property (i.e., residually finite group)
View all group property implications | View all group property non-implications
Get more facts about conjugacy-separable group|Get more facts about residually finite group

Definitions used

Conjugacy-separable group

Further information: Conjugacy-separable group

A group $G$ is termed conjugacy-separable if given any elements $x,y \in G$ that are not conjugate in $G$, there exists a normal subgroup of finite index $N$ such that the images of $x and [itex]y$ in $G/N$ are not conjugate.

Residually finite group

Further information: Residually finite group

A group $G$ is termed residually finite if given any non-identity element $x \in G$, there is a normal subgroup of finite index $N$ in $G$ such that $x$ is not in $N$.

Proof

Given: A conjugacy-separable group $G$, a non-identity element $x \in G$.

To prove: There is a normal subgroup of finite index $N$ in $G$ such that $x$ is not in $N$.

Proof: Let $e$ be the identity element. Since no non-identity element is conjugate to the identity element, $x$ and $e$ are in distinct conjugacy classes. Thus, there exists a normal subgroup $N$ of finite index such that the image of $x$ is not conjugate to the image of $e$. In particular, this implies that $x$ is not contained in $N$, so we are done.